Complex division, imag part

Percentage Accurate: 61.7% → 82.8%

Time: 3.4s

Alternatives: 9

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-109}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (fma c c (* d d)))))
   (if (<= d -1.1e+135)
     (/ (fma b (/ c d) (- a)) d)
     (if (<= d -3e-79)
       t_0
       (if (<= d 2.05e-109)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 8e+87) t_0 (fma (/ b d) (/ c d) (/ (- a) d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / fma(c, c, (d * d));
	double tmp;
	if (d <= -1.1e+135) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (d <= -3e-79) {
		tmp = t_0;
	} else if (d <= 2.05e-109) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 8e+87) {
		tmp = t_0;
	} else {
		tmp = fma((b / d), (c / d), (-a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (d <= -1.1e+135)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (d <= -3e-79)
		tmp = t_0;
	elseif (d <= 2.05e-109)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 8e+87)
		tmp = t_0;
	else
		tmp = fma(Float64(b / d), Float64(c / d), Float64(Float64(-a) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e+135], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3e-79], t$95$0, If[LessEqual[d, 2.05e-109], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 8e+87], t$95$0, N[(N[(b / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.1e135

   1. Initial program 33.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 85.9

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   if -1.1e135 < d < -3e-79 or 2.0500000000000001e-109 < d < 7.9999999999999997e87

   1. Initial program 75.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c + d \cdot d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c} + d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{d \cdot d}} \]

      4. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{{d}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, {d}^{2}\right)}} \]

      6. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

      7. lift-*.f64 75.0

         \[\leadsto \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

   3. Applied rewrites 75.0%

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   if -3e-79 < d < 2.0500000000000001e-109

   1. Initial program 71.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{b + -1 \cdot \frac{a \cdot d}{c}}{\color{blue}{c}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot d}{c}}{c} \]

      3. metadata-eval N/A

         \[\leadsto \frac{b - 1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{b - \frac{-1}{-1} \cdot \frac{a \cdot d}{c}}{c} \]

      5. times-frac N/A

         \[\leadsto \frac{b - \frac{-1 \cdot \left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      8. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      11. *-commutative N/A

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      12. lower-*.f64 89.2

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]

   if 7.9999999999999997e87 < d

   1. Initial program 40.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \color{blue}{-1 \cdot \frac{a}{d}} \]

      2. pow2 N/A

         \[\leadsto \frac{b \cdot c}{d \cdot d} + -1 \cdot \frac{a}{d} \]

      3. times-frac N/A

         \[\leadsto \frac{b}{d} \cdot \frac{c}{d} + \color{blue}{-1} \cdot \frac{a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \color{blue}{\frac{c}{d}}, -1 \cdot \frac{a}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \frac{\color{blue}{c}}{d}, -1 \cdot \frac{a}{d}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \frac{c}{\color{blue}{d}}, -1 \cdot \frac{a}{d}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-1 \cdot a}{d}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-1 \cdot a}{d}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{\mathsf{neg}\left(a\right)}{d}\right) \]

      10. lower-neg.f64 82.6

          \[\leadsto \mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-a}{d}\right) \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-a}{d}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-109}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (fma c c (* d d)))))
   (if (<= d -1.1e+135)
     (/ (fma b (/ c d) (- a)) d)
     (if (<= d -3e-79)
       t_0
       (if (<= d 2.05e-109)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 6e+87) t_0 (/ (fma c (/ b d) (- a)) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / fma(c, c, (d * d));
	double tmp;
	if (d <= -1.1e+135) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (d <= -3e-79) {
		tmp = t_0;
	} else if (d <= 2.05e-109) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 6e+87) {
		tmp = t_0;
	} else {
		tmp = fma(c, (b / d), -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (d <= -1.1e+135)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (d <= -3e-79)
		tmp = t_0;
	elseif (d <= 2.05e-109)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 6e+87)
		tmp = t_0;
	else
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e+135], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3e-79], t$95$0, If[LessEqual[d, 2.05e-109], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6e+87], t$95$0, N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.1e135

   1. Initial program 33.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 85.9

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]

   if -1.1e135 < d < -3e-79 or 2.0500000000000001e-109 < d < 5.9999999999999998e87

   1. Initial program 75.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c + d \cdot d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c} + d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{d \cdot d}} \]

      4. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{{d}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, {d}^{2}\right)}} \]

      6. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

      7. lift-*.f64 75.0

         \[\leadsto \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

   3. Applied rewrites 75.0%

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   if -3e-79 < d < 2.0500000000000001e-109

   1. Initial program 71.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{b + -1 \cdot \frac{a \cdot d}{c}}{\color{blue}{c}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot d}{c}}{c} \]

      3. metadata-eval N/A

         \[\leadsto \frac{b - 1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{b - \frac{-1}{-1} \cdot \frac{a \cdot d}{c}}{c} \]

      5. times-frac N/A

         \[\leadsto \frac{b - \frac{-1 \cdot \left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      8. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      11. *-commutative N/A

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      12. lower-*.f64 89.2

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]

   if 5.9999999999999998e87 < d

   1. Initial program 40.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c + d \cdot d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{c \cdot c} + d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{d \cdot d}} \]

      4. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{c \cdot c + \color{blue}{{d}^{2}}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, {d}^{2}\right)}} \]

      6. pow2 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

      7. lift-*.f64 40.6

         \[\leadsto \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

   3. Applied rewrites 40.6%

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

   4. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{c \cdot b}{d} + -1 \cdot a}{d} \]

      4. associate-/l* N/A

         \[\leadsto \frac{c \cdot \frac{b}{d} + -1 \cdot a}{d} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, -1 \cdot a\right)}{d} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, -1 \cdot a\right)}{d} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      8. lift-neg.f64 83.0

         \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d} \]

   6. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -2.65 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-108}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+111}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -2.65e+95)
     t_0
     (if (<= d -1.45e-43)
       (/ (fma (- a) d (* c b)) (* d d))
       (if (<= d 1.46e-108)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 3.2e+111) (* (- a) (/ d (fma d d (* c c)))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -2.65e+95) {
		tmp = t_0;
	} else if (d <= -1.45e-43) {
		tmp = fma(-a, d, (c * b)) / (d * d);
	} else if (d <= 1.46e-108) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 3.2e+111) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -2.65e+95)
		tmp = t_0;
	elseif (d <= -1.45e-43)
		tmp = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d));
	elseif (d <= 1.46e-108)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 3.2e+111)
		tmp = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -2.65e+95], t$95$0, If[LessEqual[d, -1.45e-43], N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.46e-108], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.2e+111], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.6500000000000001e95 or 3.2000000000000001e111 < d

   1. Initial program 38.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      4. lower-neg.f64 73.6

         \[\leadsto \frac{-a}{d} \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if -2.6500000000000001e95 < d < -1.4500000000000001e-43

   1. Initial program 74.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      4. +-commutative N/A

         \[\leadsto -\frac{-1 \cdot \frac{b \cdot c}{d} + a}{d} \]

      5. *-commutative N/A

         \[\leadsto -\frac{\frac{b \cdot c}{d} \cdot -1 + a}{d} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      8. *-commutative N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

      9. lower-*.f64 56.6

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

   4. Applied rewrites 56.6%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - -1 \cdot \left(b \cdot c\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(-1 \cdot b\right) \cdot c}{{d}^{2}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot c}{{d}^{2}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{2}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{\color{blue}{2}}} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      12. +-commutative N/A

          \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d + b \cdot c}{{d}^{2}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-1 \cdot a, d, b \cdot c\right)}{{d}^{2}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(a\right), d, b \cdot c\right)}{{d}^{2}} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{{d}^{2}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      19. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

      20. lift-*.f64 53.1

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

   7. Applied rewrites 53.1%

      \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\color{blue}{d \cdot d}} \]

   if -1.4500000000000001e-43 < d < 1.4600000000000001e-108

   1. Initial program 72.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{b + -1 \cdot \frac{a \cdot d}{c}}{\color{blue}{c}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot d}{c}}{c} \]

      3. metadata-eval N/A

         \[\leadsto \frac{b - 1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{b - \frac{-1}{-1} \cdot \frac{a \cdot d}{c}}{c} \]

      5. times-frac N/A

         \[\leadsto \frac{b - \frac{-1 \cdot \left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      8. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      11. *-commutative N/A

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      12. lower-*.f64 87.1

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

   4. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]

   if 1.4600000000000001e-108 < d < 3.2000000000000001e111

   1. Initial program 75.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2} + {d}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right)}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right)}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{{c}^{2}} + {d}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{{c}^{2}} + {d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(a\right)\right) \cdot d}{{\color{blue}{c}}^{2} + {d}^{2}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{{\color{blue}{c}}^{2} + {d}^{2}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{d \cdot d + {\color{blue}{c}}^{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      11. lift-*.f64 48.6

          \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

   4. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(a\right)\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(\color{blue}{d}, d, c \cdot c\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{d \cdot d + \color{blue}{c \cdot c}} \]

      7. associate-/l* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\frac{d}{d \cdot d + c \cdot c}} \]

      8. pow2 N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{d}^{2} + \color{blue}{c} \cdot c} \]

      9. pow2 N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{d}^{2} + {c}^{\color{blue}{2}}} \]

      10. +-commutative N/A

          \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{c}^{2} + \color{blue}{{d}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\frac{d}{{c}^{2} + {d}^{2}}} \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \frac{\color{blue}{d}}{{c}^{2} + {d}^{2}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{\color{blue}{d}}{{c}^{2} + {d}^{2}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      15. +-commutative N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      16. pow2 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{d \cdot d + {\color{blue}{c}}^{2}} \]

      17. pow2 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{d \cdot d + c \cdot \color{blue}{c}} \]

      18. lift-fma.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, \color{blue}{d}, c \cdot c\right)} \]

      19. lift-*.f64 52.7

          \[\leadsto \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

   6. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -2.65 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-108}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+111}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -2.65e+95)
     t_0
     (if (<= d -1.45e-43)
       (/ (fma (- a) d (* c b)) (* d d))
       (if (<= d 1.46e-108)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 3.2e+111) (* (- a) (/ d (fma d d (* c c)))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -2.65e+95) {
		tmp = t_0;
	} else if (d <= -1.45e-43) {
		tmp = fma(-a, d, (c * b)) / (d * d);
	} else if (d <= 1.46e-108) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 3.2e+111) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -2.65e+95)
		tmp = t_0;
	elseif (d <= -1.45e-43)
		tmp = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d));
	elseif (d <= 1.46e-108)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 3.2e+111)
		tmp = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -2.65e+95], t$95$0, If[LessEqual[d, -1.45e-43], N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.46e-108], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.2e+111], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.6500000000000001e95 or 3.2000000000000001e111 < d

   1. Initial program 38.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      4. lower-neg.f64 73.6

         \[\leadsto \frac{-a}{d} \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if -2.6500000000000001e95 < d < -1.4500000000000001e-43

   1. Initial program 74.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      4. +-commutative N/A

         \[\leadsto -\frac{-1 \cdot \frac{b \cdot c}{d} + a}{d} \]

      5. *-commutative N/A

         \[\leadsto -\frac{\frac{b \cdot c}{d} \cdot -1 + a}{d} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      8. *-commutative N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

      9. lower-*.f64 56.6

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

   4. Applied rewrites 56.6%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - -1 \cdot \left(b \cdot c\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(-1 \cdot b\right) \cdot c}{{d}^{2}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot c}{{d}^{2}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{2}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{\color{blue}{2}}} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      12. +-commutative N/A

          \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d + b \cdot c}{{d}^{2}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-1 \cdot a, d, b \cdot c\right)}{{d}^{2}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(a\right), d, b \cdot c\right)}{{d}^{2}} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{{d}^{2}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      19. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

      20. lift-*.f64 53.1

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

   7. Applied rewrites 53.1%

      \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\color{blue}{d \cdot d}} \]

   if -1.4500000000000001e-43 < d < 1.4600000000000001e-108

   1. Initial program 72.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{b + -1 \cdot \frac{a \cdot d}{c}}{\color{blue}{c}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot d}{c}}{c} \]

      3. metadata-eval N/A

         \[\leadsto \frac{b - 1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{b - \frac{-1}{-1} \cdot \frac{a \cdot d}{c}}{c} \]

      5. times-frac N/A

         \[\leadsto \frac{b - \frac{-1 \cdot \left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      8. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      11. *-commutative N/A

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      12. lower-*.f64 87.1

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

   4. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      4. associate-/l* N/A

         \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

      6. lower-/.f64 86.9

         \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

   6. Applied rewrites 86.9%

      \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

   if 1.4600000000000001e-108 < d < 3.2000000000000001e111

   1. Initial program 75.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2} + {d}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right)}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right)}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{{c}^{2}} + {d}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{{c}^{2}} + {d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(a\right)\right) \cdot d}{{\color{blue}{c}}^{2} + {d}^{2}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{{\color{blue}{c}}^{2} + {d}^{2}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{d \cdot d + {\color{blue}{c}}^{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      11. lift-*.f64 48.6

          \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

   4. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(a\right)\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(\color{blue}{d}, d, c \cdot c\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{d \cdot d + \color{blue}{c \cdot c}} \]

      7. associate-/l* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\frac{d}{d \cdot d + c \cdot c}} \]

      8. pow2 N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{d}^{2} + \color{blue}{c} \cdot c} \]

      9. pow2 N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{d}^{2} + {c}^{\color{blue}{2}}} \]

      10. +-commutative N/A

          \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{c}^{2} + \color{blue}{{d}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\frac{d}{{c}^{2} + {d}^{2}}} \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \frac{\color{blue}{d}}{{c}^{2} + {d}^{2}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{\color{blue}{d}}{{c}^{2} + {d}^{2}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      15. +-commutative N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      16. pow2 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{d \cdot d + {\color{blue}{c}}^{2}} \]

      17. pow2 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{d \cdot d + c \cdot \color{blue}{c}} \]

      18. lift-fma.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, \color{blue}{d}, c \cdot c\right)} \]

      19. lift-*.f64 52.7

          \[\leadsto \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

   6. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -2.65 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-108}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+111}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -2.65e+95)
     t_0
     (if (<= d -1.4e-43)
       (/ (fma (- a) d (* c b)) (* d d))
       (if (<= d 1.25e-108)
         (/ b c)
         (if (<= d 3.2e+111) (* (- a) (/ d (fma d d (* c c)))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -2.65e+95) {
		tmp = t_0;
	} else if (d <= -1.4e-43) {
		tmp = fma(-a, d, (c * b)) / (d * d);
	} else if (d <= 1.25e-108) {
		tmp = b / c;
	} else if (d <= 3.2e+111) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -2.65e+95)
		tmp = t_0;
	elseif (d <= -1.4e-43)
		tmp = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d));
	elseif (d <= 1.25e-108)
		tmp = Float64(b / c);
	elseif (d <= 3.2e+111)
		tmp = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -2.65e+95], t$95$0, If[LessEqual[d, -1.4e-43], N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.25e-108], N[(b / c), $MachinePrecision], If[LessEqual[d, 3.2e+111], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.6500000000000001e95 or 3.2000000000000001e111 < d

   1. Initial program 38.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      4. lower-neg.f64 73.6

         \[\leadsto \frac{-a}{d} \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if -2.6500000000000001e95 < d < -1.3999999999999999e-43

   1. Initial program 74.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      4. +-commutative N/A

         \[\leadsto -\frac{-1 \cdot \frac{b \cdot c}{d} + a}{d} \]

      5. *-commutative N/A

         \[\leadsto -\frac{\frac{b \cdot c}{d} \cdot -1 + a}{d} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      8. *-commutative N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

      9. lower-*.f64 56.6

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

   4. Applied rewrites 56.6%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - -1 \cdot \left(b \cdot c\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(-1 \cdot b\right) \cdot c}{{d}^{2}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot c}{{d}^{2}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{2}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{\color{blue}{2}}} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      12. +-commutative N/A

          \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d + b \cdot c}{{d}^{2}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-1 \cdot a, d, b \cdot c\right)}{{d}^{2}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(a\right), d, b \cdot c\right)}{{d}^{2}} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{{d}^{2}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      19. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

      20. lift-*.f64 53.1

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

   7. Applied rewrites 53.1%

      \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\color{blue}{d \cdot d}} \]

   if -1.3999999999999999e-43 < d < 1.25e-108

   1. Initial program 72.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 70.4

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if 1.25e-108 < d < 3.2000000000000001e111

   1. Initial program 75.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2} + {d}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right)}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right)}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{{c}^{2}} + {d}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\color{blue}{{c}^{2}} + {d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(a\right)\right) \cdot d}{{\color{blue}{c}}^{2} + {d}^{2}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{{\color{blue}{c}}^{2} + {d}^{2}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{d \cdot d + {\color{blue}{c}}^{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, \color{blue}{d}, {c}^{2}\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      11. lift-*.f64 48.6

          \[\leadsto \frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

   4. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\left(-a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(-a\right) \cdot d}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(a\right)\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(\color{blue}{d}, d, c \cdot c\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d}{d \cdot d + \color{blue}{c \cdot c}} \]

      7. associate-/l* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\frac{d}{d \cdot d + c \cdot c}} \]

      8. pow2 N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{d}^{2} + \color{blue}{c} \cdot c} \]

      9. pow2 N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{d}^{2} + {c}^{\color{blue}{2}}} \]

      10. +-commutative N/A

          \[\leadsto \left(-1 \cdot a\right) \cdot \frac{d}{{c}^{2} + \color{blue}{{d}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-1 \cdot a\right) \cdot \color{blue}{\frac{d}{{c}^{2} + {d}^{2}}} \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \frac{\color{blue}{d}}{{c}^{2} + {d}^{2}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{\color{blue}{d}}{{c}^{2} + {d}^{2}} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{\color{blue}{{c}^{2} + {d}^{2}}} \]

      15. +-commutative N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{{d}^{2} + \color{blue}{{c}^{2}}} \]

      16. pow2 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{d \cdot d + {\color{blue}{c}}^{2}} \]

      17. pow2 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{d \cdot d + c \cdot \color{blue}{c}} \]

      18. lift-fma.f64 N/A

          \[\leadsto \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, \color{blue}{d}, c \cdot c\right)} \]

      19. lift-*.f64 52.6

          \[\leadsto \left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \]

   6. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{if}\;d \leq -2.65 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-108}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (fma (- a) d (* c b)) (* d d))))
   (if (<= d -2.65e+95)
     t_0
     (if (<= d -1.4e-43)
       t_1
       (if (<= d 1.46e-108) (/ b c) (if (<= d 2e+95) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = fma(-a, d, (c * b)) / (d * d);
	double tmp;
	if (d <= -2.65e+95) {
		tmp = t_0;
	} else if (d <= -1.4e-43) {
		tmp = t_1;
	} else if (d <= 1.46e-108) {
		tmp = b / c;
	} else if (d <= 2e+95) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d))
	tmp = 0.0
	if (d <= -2.65e+95)
		tmp = t_0;
	elseif (d <= -1.4e-43)
		tmp = t_1;
	elseif (d <= 1.46e-108)
		tmp = Float64(b / c);
	elseif (d <= 2e+95)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.65e+95], t$95$0, If[LessEqual[d, -1.4e-43], t$95$1, If[LessEqual[d, 1.46e-108], N[(b / c), $MachinePrecision], If[LessEqual[d, 2e+95], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.6500000000000001e95 or 2.00000000000000004e95 < d

   1. Initial program 39.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      4. lower-neg.f64 72.6

         \[\leadsto \frac{-a}{d} \]

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if -2.6500000000000001e95 < d < -1.3999999999999999e-43 or 1.4600000000000001e-108 < d < 2.00000000000000004e95

   1. Initial program 75.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{a + -1 \cdot \frac{b \cdot c}{d}}{d} \]

      4. +-commutative N/A

         \[\leadsto -\frac{-1 \cdot \frac{b \cdot c}{d} + a}{d} \]

      5. *-commutative N/A

         \[\leadsto -\frac{\frac{b \cdot c}{d} \cdot -1 + a}{d} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{b \cdot c}{d}, -1, a\right)}{d} \]

      8. *-commutative N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

      9. lower-*.f64 53.3

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d} \]

   4. Applied rewrites 53.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{c \cdot b}{d}, -1, a\right)}{d}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - -1 \cdot \left(b \cdot c\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(-1 \cdot b\right) \cdot c}{{d}^{2}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot c}{{d}^{2}} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{2}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{b \cdot c - a \cdot d}{{d}^{\color{blue}{2}}} \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a\right)\right) \cdot d}{{d}^{2}} \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \frac{b \cdot c + \left(\mathsf{neg}\left(a \cdot d\right)\right)}{{d}^{2}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{b \cdot c + -1 \cdot \left(a \cdot d\right)}{{d}^{2}} \]

      12. +-commutative N/A

          \[\leadsto \frac{-1 \cdot \left(a \cdot d\right) + b \cdot c}{{d}^{2}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(-1 \cdot a\right) \cdot d + b \cdot c}{{d}^{2}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-1 \cdot a, d, b \cdot c\right)}{{d}^{2}} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(a\right), d, b \cdot c\right)}{{d}^{2}} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{{d}^{2}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{{d}^{2}} \]

      19. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

      20. lift-*.f64 49.9

          \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d} \]

   7. Applied rewrites 49.9%

      \[\leadsto \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\color{blue}{d \cdot d}} \]

   if -1.3999999999999999e-43 < d < 1.4600000000000001e-108

   1. Initial program 72.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 70.5

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -2.15e-42)
     t_0
     (if (<= c 1.85e+33) (/ (fma b (/ c d) (- a)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -2.15e-42) {
		tmp = t_0;
	} else if (c <= 1.85e+33) {
		tmp = fma(b, (c / d), -a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -2.15e-42)
		tmp = t_0;
	elseif (c <= 1.85e+33)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.15e-42], t$95$0, If[LessEqual[c, 1.85e+33], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.1500000000000001e-42 or 1.8499999999999999e33 < c

   1. Initial program 50.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{b + -1 \cdot \frac{a \cdot d}{c}}{\color{blue}{c}} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot d}{c}}{c} \]

      3. metadata-eval N/A

         \[\leadsto \frac{b - 1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. metadata-eval N/A

         \[\leadsto \frac{b - \frac{-1}{-1} \cdot \frac{a \cdot d}{c}}{c} \]

      5. times-frac N/A

         \[\leadsto \frac{b - \frac{-1 \cdot \left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{-1 \cdot c}}{c} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      8. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      11. *-commutative N/A

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      12. lower-*.f64 71.4

          \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{b - \frac{d \cdot a}{c}}{c}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      4. associate-/l* N/A

         \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

      6. lower-/.f64 75.3

         \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

   6. Applied rewrites 75.3%

      \[\leadsto \frac{b - a \cdot \frac{d}{c}}{c} \]

   if -2.1500000000000001e-42 < c < 1.8499999999999999e33

   1. Initial program 73.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{b \cdot c}{d} + -1 \cdot a}{d} \]

      3. associate-/l* N/A

         \[\leadsto \frac{b \cdot \frac{c}{d} + -1 \cdot a}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -1 \cdot a\right)}{d} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, \mathsf{neg}\left(a\right)\right)}{d} \]

      7. lower-neg.f64 80.9

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d} \]

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.46 \cdot 10^{-108}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -4.4e-17) t_0 (if (<= d 1.46e-108) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -4.4e-17) {
		tmp = t_0;
	} else if (d <= 1.46e-108) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-4.4d-17)) then
        tmp = t_0
    else if (d <= 1.46d-108) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -4.4e-17) {
		tmp = t_0;
	} else if (d <= 1.46e-108) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -4.4e-17:
		tmp = t_0
	elif d <= 1.46e-108:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -4.4e-17)
		tmp = t_0;
	elseif (d <= 1.46e-108)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -4.4e-17)
		tmp = t_0;
	elseif (d <= 1.46e-108)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -4.4e-17], t$95$0, If[LessEqual[d, 1.46e-108], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.4e-17 or 1.4600000000000001e-108 < d

   1. Initial program 54.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{\color{blue}{d}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      4. lower-neg.f64 58.9

         \[\leadsto \frac{-a}{d} \]

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if -4.4e-17 < d < 1.4600000000000001e-108

   1. Initial program 72.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 69.2

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{c} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ b c))

C

double code(double a, double b, double c, double d) {
	return b / c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = b / c
end function

Java

public static double code(double a, double b, double c, double d) {
	return b / c;
}

Python

def code(a, b, c, d):
	return b / c

Julia

function code(a, b, c, d)
	return Float64(b / c)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = b / c;
end

Wolfram

code[a_, b_, c_, d_] := N[(b / c), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Step-by-step derivation

   1. lower-/.f64 43.2

      \[\leadsto \frac{b}{\color{blue}{c}} \]

4. Applied rewrites 43.2%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
5. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Less[N[Abs[d], $MachinePrecision], N[Abs[c], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025106 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))